# Common Nets of Boxes

• Ryuhei Uehara
Chapter

## Abstract

In this chapter, we first consider problems that are easy to handle with computers. Since it seems to be hard to deal with the real coordinate system, polygons on a square grid would be reasonable. Speaking of polyhedra that can be folded from a polygon on a square gird, the first thing that comes to mind is a rectangular parallelepiped, or “box”. Is there a single polygon on a square grid that can be folded into multiple rectangular parallelepipeds? The answer is [Yes]. In 1999, Biedl et al. succeeded in actually finding out two polygons that satisfy the following property: “each of two can fold into two different types of boxes” [BCD+99]. (The actual examples are shown in [DO07, Fig. 25.53]. Looking at these two examples, it is hard to make it out that the boxes are folded from it. They said that they found out through trial and error without using a computer.) An example found by me is shown in Fig. 3.1. This example is one of the easiest to understand (by my subjectivity). Are these polygons “exceptional” ones with only a few? Actually, it is not. In this chapter, we introduce the latest results on nets in which these multiple boxes can be folded. Many interesting nets brought by combinations of mathematics and algorithms appear. If you are good at programming, you may want to make it yourself and make new discoveries.

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